Properties of Equality
by ekc293
Summary: "It was essentially the basic Properties of Equality. Her dad was important to Kate; Alexis was important to her Dad; therefore, Alexis was important to Kate." Two-shot, Kate x Alexis family fluff with some Caskett to taste set sometime after 04x07.
1. Thank You

GUESS WHO FINISHED HER FIRST SEMESTER OF COLLEGE GUYS? WHOOOOOOOOO!  
>And in celebration here's a new story!<br>A big thank you to anyone who has reviewed/favorited any of my stories recently! I really appreciate it more than you know :)

Anyway... this is my version of how Alexis realizes that Kate loves her father and is not some she-demon hell-bent on destroying her life.

Read on, my friends!

**Disclaimer** - I don't own Castle. If Andrew Marlowes's writing ability was a baseball, it would be a 583 ft homerun. Mine, on the other hand, would be a bunt. (I read over it to check for mistakes. There still may be some. And by "may be" I really mean probably, so I apologize in advance)

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><p>Before Alexis even opened her eyes, the first thing she was aware of was the sound of beeping. The sound was a steady beeping that, now that she focused on it, coincided with the sound of her own heartbeat. She tried to take a deep breath but there was something in her nose that made that nearly impossible. Not to mention it made her nose tickle.<p>

Opening her eyes slowly, Alexis quickly shut them again, trying to block out the bright light that was on the other side of her eyelids.

"Alexis?" she heard a voice whisper.

Groaning slightly, Alexis turned her head and cracked her eyes open, taking in the image of the person sitting next to her bed.

The last person she expected to be sitting next to her, however, was the one and only Detective Beckett.

Seeing the girl's eyes open, Beckett gave Alexis a genuine smile and stood up. Pressing a button on the table beside her quickly, Beckett moved to stand directly next to Alexis's head.

"Hey Alexis," the Detective said quietly, "how are you feeling?"

Suddenly as she became more aware of her surroundings, Alexis's nose started to feel cold and uncomfortable. Moving her hand up to remove the annoyance, she was surprised to feel Beckett's hand wrap around her wrist, moving it back down to her side.

"I don't think that's such a good idea, sweetie." Kate said, laying her hand by her side. Then Kate took her other hand and ran it slowly across Alexis's hairline, brushing some fly-away strands away from her face, "the doctor will want you to keep that in until he sees you."

Alexis's brow furrowed and Beckett, realizing what she was doing, quickly removed her hand and took a small step back from the bed. She took a deep breath.

"You gave us quite a scare back there, Alexis."

It was then that Alexis finally realized where she was. The beeping machine, the cold air up her nose, the uncomfortable bed and the presence of the Detective could only mean one thing.

She was hooked up to a machine, hooked up on oxygen, in a hospital.

How did she end up here?

"What happened?" Alexis croaked. She coughed a little to clear her throat.

She watched as the Detective's brow furrowed, "Do you remember anything?"

Alexis thought back to earlier in the day. She had gotten up at 6:15 like she did every morning, showered, got dressed for school, ate a quick breakfast, left her Dad a note telling him when she'd be back (because he was still asleep), grabbed her keys, and walked out of the loft. She had gotten to school early, talked to Paige for awhile, and then went to class. She took her English exam, and there had been a pop-quiz in Art History, and…

No matter how hard she tried, Alexis couldn't remember anything else.

"I remember… getting up and leaving… leaving the loft. I… got to school early and I talked to Paige. I had an exam in my English class and… and then I remember a pop-quiz in Art History, but… But I don't remember anything else." She admitted.

Kate gave Alexis a sad smile and picked up where Alexis left off.

"You were walking in the hallway with your friends and when you got to your locker, you fainted."

The heartbeat monitor started beating faster, and Alexis's eyes got bigger. "What? I fainted? How?"

"Dehydration," the older woman said simply, "the doctors think that you've been doing too much."

"But why are you here?" As soon as Alexis said it, she wished she could take it back. Or at least make it sound like it wasn't an accusation.

It wasn't that she hated the woman. Really, that wasn't it at all. If anything, she respected the Detective more than any other woman in her life. What she didn't respect was what she was doing to her father. She saw how devastated her father could become because of Detective Kate Beckett. She saw that her Dad was hurting and Katherine Beckett was the reason, and that wasn't okay with her because no matter what happened or whose fault it was or how much she respected the Detective her Dad was still the most important person in her life. Alexis would rather die than see him hurt or worse because of this woman who has slowly become such an integral part of his heart.

Alexis watched as Beckett shifted her weight awkwardly from one foot to another as she tried to explain.

"The hospital tried to get in contact with your father, but he has his phone turned off for his meeting and all of the lines at Black Pawn are down today because they're changing service providers. They tried calling your grandmother but her phone went straight to voicemail," she shrugged her shoulders, "your father apparently put me down as your third emergency contact. The hospital called me, and I came."

Alexis nodded shortly, and looked as if she was about to speak, when she heard the door open. Both women turned towards the sound and watched as a man wearing scrubs and a lab jacket walk in towards her bed.

"Hello, Miss Castle, I'm Doctor Alvarez," the man grabbed her chart from the edge of the bed and quickly flipped through the pages, "it looks to me like you've been burning both ends of the candle." He gave Alexis a quick smile, "let's get you checked out so we can get you out of here as quickly as possible okay?"

Doctor Alvarez began to move around the bed and started to check Alexis's vitals, recording what was important and telling her what was going on with her body and about what she would have to do to get better.

Alexis was only partially paying attention though, because she kept sneaking glances at the Detective who was still in the room. Beckett had stepped back away from the bed to make room for the doctor. She had crossed her arms over her chest, but her eyes followed every move the doctor made, listening to every word he said about her condition.

"Well, Miss Castle," Doctor Alvarez said, "Once we get some fluids into you, you'll be as good as new. We're going to keep you for overnight observation. We'll try to get in contact with your father again…"

Kate interrupted the doctor, "That won't be necessary, Doctor Alvarez. I got in contact with her father. He's on his way now."

Doctor Alvarez nodded his head in acknowledgment, "Alright then, Miss Castle. Don't try to move out of bed, and if you need anything, just press this button." Doctor Alvarez picked up the button that Kate had pressed earlier of the table beside her and put it within reaching distance. "I'll be back to check up on you later this afternoon." And with that, Doctor Alvarez left the two women alone.

"My Dad is coming?" Alexis said quietly, once she heard the door close.

Beckett moved closer to the bed again, nodding her head. "Yeah, he called a couple minutes before you woke up and I told him I was here with you. He should be here any minute."

"Were you in the middle of a case?"

The Detective nodded again.

"I'm sorry," Alexis said, "I would've been alright if you hadn't come. I know you're busy."

Beckett smiled at the young girl and shrugged her shoulders, "It's nothing Ryan and Esposito can't handle. They're big boys."

Alexis took in the appearance of the older woman. She was wearing a structured blazer, her badge and piece visible on her hip. Her hair fell across her shoulders and she was wearing what her dad had called her "power heels". However, Alexis also noticed the shadows underneath her eyes. The woman's exhaustion was visible in the slight slump in her shoulders. But there was a tenderness in her eyes that Alexis only saw on rare occasions.

Alexis felt a quick flash of guilt sink into the pit of her stomach. This woman worked so hard at her job and now she was being forced to look after her as well. Every time Alexis saw Beckett since that day outside of the bank, the Detective had just been progressively looking more and more tired.

That day outside the bank. The hostage situation. Her break-up with Ashley. Alexis yelling at the person who was responsible for getting her father out of that bank alive. The same person who was standing right beside her now.

"Detective…"

"Alexis, please call me Kate."

"I never did get to actually thank you," Alexis mumbled quietly, almost embarrassed, "for getting Dad and Grams out of that bank."

Before Alexis was even finished speaking, "You don't need to thank me, Alexis. I was just doing my job."

This time it was Alexis's turn to shake her head. "No, you're a homicide detective. Your job is not to save hostages, but…"

Kate cut her off gently, "Alexis, above anything else, my job is to protect people. If that means getting killers off the street, I'll do it. If that means hostage negotiation, I'll do it. And the minute I realized your Dad was in there, there wasn't a single thing anyone could do to keep me from getting involved. It's my job, to protect him, Alexis, and I _will_ do anything necessary to keep him safe."

By the end of her speech, Alexis was breathless at the fierceness of the determination of the Detective's words. Her words were angry, as if the thought of someone harming her father was enough to make her physically violent. They were passionate, as if the protection of her father was the most important thing in her life.

When Alexis looked into the Detective's eyes, however, she was blown away by what she saw. It was then Alexis realized the real reason Kate was here in this hospital room with her. Kate cared about her father. Kate came as soon as she got the call because she knew how important Alexis was to him.

It was essentially the basic Properties of Equality.

Her dad was important to Kate; Alexis was important to her Dad; therefore, Alexis was important to Kate.

Kate would do anything for her father; her father would do anything for Alexis; therefore, Kate would do anything for Alexis.

Alexis looked at Kate, and the Detective saw the epiphany flash across her face. Kate blushed slightly.

Alexis opened her mouth to speak multiple times, but nothing came out. Eventually, she managed to form a sentence.

"Kate," she said quietly, trying to keep her voice under control, "I -"

"Alexis!"

The girl in the hospital quickly turned her head to the familiar voice, not surprised at all to see her father running into the hospital room, his tie hanging loose around his neck and his hair standing up at odd angles like he had been running his hands through it. He quickly found his way to the edge of her bed. Grabbing her hand in one of his, he let his other hand drift over her hair, making sure that every single one was still there. Then he leaned down and kissed her softly on the forehead.

"I'm so happy you're alright sweetheart."

Alexis gave him a smile and squeezed his hand, "I'm glad you're here, Dad."

While Rick kept his running his hand across his daughter's hair, Alexis tried to find Kate. She wasn't where she had been standing earlier. Glancing around the edge of the room, she finally made eye contact with the Detective as she was sneaking out the door. Alexis opened her mouth to call her back, but stopped when she saw the look on Kate's face. It was the exact same smile she had given her the day when Kate had gotten her Dad and Grams safely out of the bank: the smile that said everything that the Detective wasn't ready to put into words yet. Before Alexis could respond however, the door clicked shut and the Detective was gone.

It was then that Alexis realized her father was still talking.

"I'm so sorry that I wasn't here sooner. My phone was turned off because of the meeting and if I would've known what was happening I wouldn't have left you. I got here as soon as I could. Every morning I'm going to make you breakfast. I'm going to make sure that you start eating better. You're going off to college soon, sweetie. You need to take better care of yourself. When I got that text from Detective Beckett… Wait? Where is Kate?"

Alexis raised an eyebrow at her father, "She just left. She was standing right there," Alexis pointed to the spot just behind the chair, "when you came in."

Her father groaned and pulled out his phone, intent on calling his Detective and thanking her profusely for being there for his daughter when he wasn't.

Alexis didn't shy away from calling Kate "his Detective" because that was exactly what she was. She saw that now. She was his detective and he was her writer and as unorthodox as their relationship was they were partners. She would do anything for him the same way he would do anything for her.

Alexis plucked the phone from her father's hands just before he hit the dial button.

"It's okay, Dad," she whispered, "she knows."

Rick looked like he was about to protest, but Alexis cut him off.

"Thank her in person the next time you see her."

He nodded and settled into the chair beside her bed, "You should get some sleep, pumpkin. It'll help you get better."

Alexis nodded, suddenly feeling exhausted, and tried to get comfortable in the hospital bed. She squeezed her Dad's hand once and shut her eyes, but her mind kept formulating equations.

_Dad is important to Kate; I'm important to my dad; therefore I'm important to Kate._

_Dad is important to me; Kate is important to my dad; therefore Kate is important to me._

Alexis smiled.

"Dad?" she mumbled, her eyes still closed.

"Yes, pumpkin?"

"Can you do me a favor?"

"Of course, Alexis. Anything you need."

Alexis snuggled further into the pillows, trying to hide the fact that her smile had transformed into a smirk.

"The next time you see Kate," she whispered, "can you give her a kiss for me? Tell her I said thank you. For everything."

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><p>It is my honest opinion that Alexis is secretly the biggest Caskett shipper of them all.<br>**Love it? Hate it? Let me know what you think!**


	2. For Everything

I know some of you expressed interest in wanting to see the ending of the original one-shot explored.  
>so... yeah. This happened.<p>

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><p>After Kate left Rick with Alexis at the hospital she returned to the precinct, to catch up on what she had missed during the case. After assuring Ryan and Esposito that Alexis was awake and she would be fine (they really did love her like they were her uncles) they got to work trying to catch their killer. Castle didn't call and she didn't expect him to.<p>

In fact, Kate didn't expect to see Castle for the next few days. She knew how important Alexis was to him and sincerely wanted him to be there with his daughter. She would probably kick him out and tell him to go home if he showed up.

That didn't stop him from missing him though.

Their case ended up being a relatively easy to solve; the killer left his prints all over the gun, which he failed to dispose of properly, and confessed only minutes after Beckett joined him in interrogation room two.

So the afternoon after her hospital adventure with Alexis, she found herself sitting alone at her desk with a stack of files from a case long since cold on the surface in front of her, looking at the one she held in her hands.

When she heard the elevator ding open, she didn't even bother to look up and see who it was. It wasn't until the familiar aroma of her usual coffee creeped into her nose and she saw the a large hand placing the take-out mug in front of her that she looked up from the file.

"Good afternoon, Detective," Castle said with a smile, easing into his chair by her desk.

"What are you doing here?" she detected, suspicion lacing her words.

"What?" Castle exclaimed, looking affronted, "Can't a man want to spend time with his partner?"

Beckett raised an eyebrow. Castle gave in, avoiding eye contact.

"Alexis asked me to leave."

When he looked up at his detective however, he realized she didn't believe him. At all. He sighed dramatically.

"Fine, she kicked me out. Told me to, and I quote, 'go pester someone else.'"

Kate let out a small laugh, closing the file and putting it on her desk, "So you came to me?"

Castle shrugged, his school boy smile making its way across his face, "There's no one I'd rather pester… although I'd like to believe that I do more than _pester_ you."

Kate rolled her eyes and shook her head, but the smile on her face made it clear that she was amused.

And she didn't deny it. It was enough to keep the smile on his face for the rest of the day. Perhaps even for the rest of the week.

"So Alexis is alright?" Kate asked, interrupting his thoughts, concern coloring her words, "She's rehydrated and as good as new?"

As if his heart couldn't be more ready to burst.

Kate saw the way his eyes softened at the mention of his daughter, but when his entire expression shifted from teasing happiness to content thoughtfulness, she knew it had everything to do with her. She felt her cheeks heat up slightly, and she moved her gaze to the armrest of his chair to avoid his eyes.

She noticed him sit up a little straighter, all the while still looking at her.

"Yes, she's fine. They released her this morning after being hooked up to an IV all night getting hydrated again. She's angry that she had to miss school today. Finals are coming up and she doesn't want to miss a single day of studying. Hence, the reason I got kicked out of the loft this afternoon." Rick shook his head in amusement before lifting his eyes back to the detective. "But yes, Kate, Alexis is fine."

The emotion in his eyes almost made her look away, but she forced herself to keep eye contact this time. She smiled.

"Good, I'm glad she's alright."

He nodded, knowing that there was more to it. Rubbing his lips together, he took a deep breath.

"Thank you," he said.

She smiled and moved her chair back to its original spot and picked up the file that she had put down when he came in. "There's no need to thank me, Castle."

Before she could even register what was happening, Castle was up out of his chair and kneeling next to her. Trying to ignore the stares of the other detectives and what this situation would look like to an outside intruder, Kate looked at Rick with eyebrows raised, silently telling him to get off the ground and back to his seat or she was going to make sure he couldn't sit down for the next week.

If he saw the threat, he ignored it. In fact, he moved one of his hands and gently clasped it around her wrist, the other sneaking through the air between them and plucking the file from her fingers placing it on the desk in front of her without ever moving his eyes from her. Then, with her hands free, he moved his hands from her wrist and wrapped her fingers in his own.

"No," he said earnestly, "I do. You were there for Alexis when I couldn't be, and you really didn't have to."

She gave him a small, uncomfortable smile, shaking her head at him, fully aware of all of the eyes on them at that moment.

"Of course I had to, Rick. She's your daughter," she took a deep breath, her voice lowering to an almost imperceptible level, "she's important to you and you are important to me. Therefore, Alexis is important to me, too."

Perhaps it was the holiday season that was letting him get away with all of this, in the middle of the precinct no less. Maybe she was just feeling especially flirty that door. Or it may have even been the look that he was giving her. Whatever the reason, Castle found that he didn't really care what the reason was. He was going to take advantage of it, even if his vision was slightly blurred by the fact that she just willingly admitted that his family and their wellbeing was important to her.

"I didn't even realize you were there," he muttered, embarrassed.

Without thinking, she gave his fingers a comforting squeeze, "You had more important things to worry about."

Castle's brow furrowed. He knows this, and obviously she knew it to that his number one priority was always going to be Alexis, but this is Kate Beckett they were talking about. After his mother and his daughter, she was by far the most important thing in his life. She would always come third in his mind…

Even in his head, he knew that didn't sound good and is not the thing to say to the woman you're in love with, especially when said woman has a gun and knows exactly how to use it.

"Still…" he started.

Kate cut him off, "It's okay, Castle. I'm a big girl. You needed to take care of your girl."

He nodded, once again thankful that she understood but frustrated that he couldn't find the words to tell her that she was just as important to him as his daughter was, just in a different way. And of course it had nothing to do with the fact that he didn't know how to say that _she _was his girl too without getting physically injured in some way.

So he just stuck with what he knew how to say.

"Thank you," he said, his eyes shining with sincerity.

She smiled back at him.

"Anytime, Rick."

She watched as he smiled and squeezed her hand briefly before letting go of his grip and rocking back on his heels, standing up and moving back to his spot in his chair. She bit her lip and grabbed her coffee from its place on her desk, bringing the warm liquid to her lips with a sigh.

"Thanks for the coffee," she said, taking a sip.

"Always," he smiled.

Kate twisted in her seat and flipped open the file she had been looking at before. She grabbed another file off her desk and handed it to him, knowing he'd want something to keep him distracted.

They sat in silence looking at the files. When she saw him surreptitiously check his watch not even 5 minutes later, she choked back a laugh.

"Castle, go home and be with your daughter."

He looked appropriately abashed at getting caught.

"Are you sure you're okay here?"

Kate rolled her eyes and gestured towards her desk. "It's cold cases all day until a body drops. And contrary to popular belief, Castle, I _can _do my job without you."

Castle nodded and stood up, putting the file back on her desk and readjusted his jacket.

"You'll call me if a body drops?" he said, sounding hopeful.

She nodded, "Of course."

He nodded and gave her a small nod, saying goodbye without uttering the words. She smiled and nodded back at him watching him leave for a moment before turning back to the file on her desk. She leaned forward slightly and cradled her head in her hand as she began looking at crime scene photographs.

Kate was so focused on the file in front of her, that she didn't hear Rick's footsteps moving back towards her desk. Before she could register what was happening, she felt his lips press against her check.

She froze slightly and as quickly as his lips appeared on her skin they were gone again. He tilted his head so he could whisper in her ear.

"That was from Alexis," he said softly, "she told me to tell you thank you. For everything."

Kate tried her hardest to hide her reaction to his kiss, pursing her lips before whispering back.

"Tell her she's very welcome, and I'm here for her if she ever needs me for anything."

She could feel him nod and start to pull away.

"Will do, Detective." She could hear the smile in his voice.

Before Kate could talk herself out of it (and before he had the chance to get too far away), Kate reached out to grab his forearm and pulled him back down to her. As soon as he was close enough to her, she pushed herself up slightly in her chair and softly kissed his cheek, letting her lips linger slightly before pulling back and releasing his arm.

He was silent for a moment before he opened his eyes and cleared his throat.

"Am I passing that onto her as well?" He said, trying to control the urge to grab her face in his hands and kiss her breathless.

She pursed her lips, trying to look as though she was thinking hard about her answer. Then she shook her head. "No, that one is just for you."

He looked at her, happiness radiating from his eyes.

"Goodbye, Castle," she smiled shyly.

And as she watched him walk away, she saw him lift his fingers to touch the spot where her lips had been.

It wasn't until Esposito coughed loudly that she realized she had been doing the same thing.

Turning to look at Ryan and Esposito, she saw them staring at her, knowing grins on their faces. She felt her own heat up slightly, before she turned back to her file, grumbling at the pair.

"Get back to work."

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><p>So... there.<br>**Love it? Hate it? Let me know what you think!**


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